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This [http://users.math.cas.cz/~gavinsky/papers/QuMoClaV.pdf example protocol] is a private-key protocol which implements Quantum Money, a unique object generated by a Trusted Third Party (TTP). It is then circulated among untrusted clients (Transferability). Each client should be able to prove the authenticity of his owned quantum money to a verifier. On the other hand, an adversary must fail in counterfeiting the quantum money with overwhelmingly high probability (Unforgeability). <br>
This [http://users.math.cas.cz/~gavinsky/papers/QuMoClaV.pdf example protocol] implements Quantum Money which is a unique object generated by a Trusted Third Party (TTP). It is then circulated among untrusted clients (Transferability). Each client should be able to prove the authenticity of his owned quantum money to a verifier. On the other hand, an adversary must fail in counterfeiting the quantum money with overwhelmingly high probability (Unforgeability). <br>


'''Tags:''' [[:Category: Multi Party Protocols|Multi Party Protocols]], [[:Category: Quantum Enhanced Classical Functionality|Quantum Enhanced Classical Functionality]], [[:Category: Specific Task|Specific Task]], Prepare (bank) and Measure (client)  
'''Tags''': Multiparty, Quantum Enhanced Classical functionality, prepare (bank) and measure (client)
[[Category: Specific Tasks]]
[[Category: Quantum Enhanced Classical Functionality]]
[[Category: Multi Party Protocols]]




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* '''Quantum coin Verification''' - To verify a quantum coin through classical communication with the TTP, its holder sends the identification number of the quantum coin to the TTP. Then, the TTP and the coin holder exchange some classical information for choosing some quantum registers. The coin holder measures the chosen registers and sends their corresponding classical information to the TTP. The TTP verifies the authenticity of the coin by the secret information he possesses.
* '''Quantum coin Verification''' - To verify a quantum coin through classical communication with the TTP, its holder sends the identification number of the quantum coin to the TTP. Then, the TTP and the coin holder exchange some classical information for choosing some quantum registers. The coin holder measures the chosen registers and sends their corresponding classical information to the TTP. The TTP verifies the authenticity of the coin by the secret information he possesses.
==Notations==
==Notations==
* <math>HMP_4</math>-states: <math>|\alpha(x)\rangle=\dfrac{1}{2}\sum_{1\leq i\leq4}(-1)^{x_i}|i\rangle</math>, <math>x\in\{0, 1\}^4</math>
* <math>HMP_4</math>-states: <math>|\alpha(x)\rangle=\dfrac{1}{2}\sum_{1\leq i\leq4}(-1)^{x_i}|i\rangle</math>, <math>x\in\{0, 1\}</math>
* for <math>m, a, b \in \{0, 1\}</math>, <math>(x, m, a, b) \in HMP_4 </math> if <math> b = \begin{cases}
x_1 \oplus x_{2+m} & \text{if } a = 0 \\
x_{3-m} \oplus x_4 & \text{if } a = 1 \end{cases}</math>
 
* <math>HMP_4</math>-queries: An <math>HMP_4</math>-query is an element <math>m \in \{0, 1\}</math>. A valid answer to the query w.r.t. <math>x \in \{0, 1\}^4</math> is a pair <math>(a, b) \in \{0, 1\} \times \{0, 1\}</math>, such that <math>(x, m, a, b) \in HMP_4</math>. An <math>HMP_4</math> -state can be used to answer an <math>HMP_4</math> -query with certainty: If <math> m = 0 </math>, let
  <math> v_1 \overset{def}{=}\dfrac{|1\rangle+|2\rangle}{\sqrt{2}} </math>            <math> v_2 \overset{def}{=}\dfrac{|1\rangle-|2\rangle}{\sqrt{2}} </math>            <math> v_3 \overset{def}{=}\dfrac{|3\rangle+|4\rangle}{\sqrt{2}} </math>            <math> v_4 \overset{def}{=}\dfrac{|3\rangle-|4\rangle}{\sqrt{2}} </math>             
otherwise (m = 1), let
  <math> v_1 \overset{def}{=}\dfrac{|1\rangle+|3\rangle}{\sqrt{2}} </math>            <math> v_2 \overset{def}{=}\dfrac{|1\rangle-|3\rangle}{\sqrt{2}} </math>            <math> v_3 \overset{def}{=}\dfrac{|2\rangle+|4\rangle}{\sqrt{2}} </math>            <math> v_4 \overset{def}{=}\dfrac{|2\rangle-|4\rangle}{\sqrt{2}} </math>
 
Measure <math>|\alpha(x_i)\rangle</math> in the basis <math>{v_1, v_2, v_3, v_4}</math>, and let <math>(a, b)</math> be <math>(0, 0)</math> if the outcome is <math>v_1</math>; <math>(0, 1)</math> in the case of <math>v_2</math>; <math>(1, 0)</math> in the case of <math>v_3</math>; <math>(1, 1)</math> in the case of <math>v_4</math>. Then <math>(x, m, a, b) \in HMP_4</math> always.


==Requirements==
==Requirements==
*Network stage: [[:Category: Quantum Memory Network Stage|quantum memory network]][[Category:Quantum Memory Network Stage]].
*Network stage: [[:Category: Quantum Memory Network Stage|quantum memory network]][[Category:Quantum Memory Network Stage]].
==Knowledge Graph==
{{graph}}


== Properties ==
== Properties ==
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**Secure against an adversary who uses adaptive “attempted verifications” in order to collect information about a coin.
**Secure against an adversary who uses adaptive “attempted verifications” in order to collect information about a coin.


==Protocol Description==
== Pseudocode ==
'''Stage 1: Quantum coin generation'''<br>
'''Stage 1: Quantum coin generation'''<br>
''Input'': A secret record consists of <math>k</math> entries <math>x_1, . . . , x_k</math>,<math> x_i\in \{0,1\}^4</math><br>
''Input'': A secret record consists of <math>k</math> entries <math>x_1, . . . , x_k</math>,<math> x_i\in \{0,1\}^4</math><br>
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* The holder sends the identification number of the quantum coin to the TTP.
* The holder sends the identification number of the quantum coin to the TTP.
* The TTP chooses uniformly at random a set <math>L_{bn}\subset[k]</math> of size <math>t</math>, and sends it to the coin holder.
* The TTP chooses uniformly at random a set <math>L_{bn}\subset[k]</math> of size <math>t</math>, and sends it to the coin holder.
* The holder consults with <math>P</math> and chooses uniformly at random a set <math>L_{hl} \subset L_{bn}</math> consisting of <math>2t/3</math> yet unmarked positions. He sends <math>L_{hl}</math> to the bank and marks in <math>P</math> all the elements of <math>L_{hl}</math> as used.
* The holder consults with P and chooses uniformly at random a set <math>L_{hl} \subset L_{bn}</math> consisting of <math>2t/3</math> yet unmarked positions. He sends <math>L_{hl}</math> to the bank and marks in <math>P</math> all the elements of <math>L_{hl}</math> as used.
* The TTP chooses at random <math>2t/3</math> values <math>m_i \in\{{0, 1}\}</math>, one for each <math>i \in L_{hl}</math> , and sends them to the coin holder.
* The TTP chooses at random <math>2t/3</math> values <math>m_i \in\{{0, 1}\}</math>, one for each <math>i \in L_{hl}</math> , and sends them to the coin holder.
* The holder measures the quantum registers corresponding to the elements of <math>L_{hl}</math> in order to produce <math>2t/3</math> pairs <math>(a_i, b_i)</math> (refer to <math>HMP_4</math>-queries in Notations), such that <math>(x_i,m_i, a_i, b_i)\in HMP_4</math> for all <math>i \in L_{hl}</math>. He sends the list of <math>(a_i, b_i)</math>s to the TTP.
* The holder measures the quantum registers corresponding to the elements of <math>L_{hl}</math> in order to produce <math>2t/3</math> pairs <math>(a_i, b_i)</math>, such that <math>(x_i,m_i, a_i, b_i)\in HMP_4</math> for all <math>i \in L_{hl}</math>. He sends the list of <math>(a_i, b_i)</math>s to the TTP.
* The TTP checks whether <math>(x_i,m_i, a_i, b_i)\in HMP_4</math> for all <math>i \in L_{hl}</math>, in which case it confirms validity of the quantum coin. Otherwise, the coin is declared to be a counterfeit.
* The TTP checks whether <math>(x_i,m_i, a_i, b_i)\in HMP_4</math> for all <math>i \in L_{hl}</math>, in which case it confirms validity of the quantum coin. Otherwise, the coin is declared to be a counterfeit.


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